Proof of theorem 6.6 in Functional Analysis of Haim Brezis (Fredholm alternative)

Question

I'm reading the proof of the following theorem:

My question is in part d), the last paragraph says that:

I'm not sure why $$ N(I-T) \subseteq N(I-T^{**}) $$ because $$ N(I-T) \subseteq E \quad\mbox{and}\quad N(I-T^{**})\subseteq E'' $$